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Chapter 10 –Testing Claims Regarding a Parameter
The other branch of statistical inference is concerned with testing
hypotheses about the value of parameters.
Defn
:
A hypothesis
is a statement about the value of a population
parameter.
Defn
:
In a hypothesis test, the hypothesis which the researcher is
attempting to prove is called the alternative (or research) hypothesis
,
denoted H
a
.
The negation of the alternative hypothesis is called the
null hypothesis
, H
0
.
The null hypothesis usually represents current
belief about the value of the parameter.
The researcher suspects that
the null hypothesis is false, and wants to disprove it.
Note
:
Hypotheses are tested in pairs (here the symbol
μ
0
represents
some specified number):
If the null hypothesis is H
0
:
μ
=
μ
0
, then the corresponding
alternative hypothesis is H
a
:
μ
≠
μ
0
.
If the null hypothesis is H
0
:
μ
≥
μ
0
, then the corresponding
alternative hypothesis is H
a
:
μ
<
μ
0
.
If the null hypothesis is H
0
:
μ
≤
μ
0
, then the corresponding
alternative hypothesis is H
a
:
μ
>
μ
0
.
In each case, the alternative hypothesis is a strict inequality; the null
hypothesis is not.
The above hypothesis statements are written in
terms of a population mean,
μ
.
If we are testing hypotheses about a
population proportion, p, or some other parameter, that parameter
would appear instead of
μ
.
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:
If the researcher wants to prove that the proportion, p, of
the U.S. population who catch a cold between October 1 and March
31 is more than 40%, the two hypotheses would be stated as follows:
H
0
:
p
≤
0.40
H
a
:
p > 0.40
Example
:
If the researcher is in charge of quality control for the
PepsiCola Company, and wants to determine whether an assembly
line at a plant which produces 12oz. cans of Pepsi is working
properly, the hypotheses would be stated as follows in terms of the
mean amount of Pepsi in the cans:
H
0
:
μ
= 12 oz.
H
a
:
μ
≠
12 oz.
Note
:
We cannot prove the null hypothesis; we may disprove it, or
we may fail to disprove it.
After taking a sample and conducting a
test, we will either reject H
0
and believe H
a
, or we will fail to reject
H
0
because of insufficient evidence against it.
As Dr. Carl Sagan
once said,
“Absence of evidence is not evidence of absence.”
There are four possibilities:
1)
We reject the null hypothesis when the null hypothesis is, in fact,
true.
In this case, we make a Type I Error
.
2)
We reject the null hypothesis when the null hypothesis is, in fact,
false.
In this case, we make a correct decision.
3)
We fail to reject the null hypothesis when the null hypothesis is,
in fact, true.
In this case, we make a correct decision.
4)
We fail to reject the null hypothesis when the null hypothesis is,
in fact, false.
In this case, we make a Type II Error
.
We want to maximize the probability of making a correct decision,
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This note was uploaded on 07/28/2011 for the course STA 2014 taught by Professor Staff during the Fall '10 term at University of Florida.
 Fall '10
 Staff
 Statistics

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